Abstract

In this work, we examine the main cosmological effects derived from a time-varying coupling (¯Q) between a dark matter (DM) fluid and a dark energy (DE) fluid with
time-varying DE equation of state (EoS) parameter (ωDE), in two different coupled DE models. These scenarios were built in terms of Chebyschev polynomials.
Our results show that such models within dark sector can suffer an instability in their perturbations at early times and a slight departure on the amplitude of the cosmic
structure growth (fσ8) from the standard background evolution of the matter. These effects depend on the form of both ¯Q and ωDE. Here, we also
perform a combined statistical analysis using current data to put tighter constraints on the parameters space. Finally, we use some selections criteria to distinguish our
models.

Ii Background equations of motion

We consider here a flat Friedmann-Robertson-Walker (FRW) universe composed by radiation, baryons, DM and DE. Moreover, we postulate that the dark components can also
interact through a non-gravitational coupling ¯Q. For ¯Q>0 (¯Q<0) implies that the energy flows from DE to DM (the energy flows from DM to DE).
Also, to satisfy the requirements imposed by local gravity experiments Koyama2009-Brax2010 () we also assume that baryons
and radiation are coupled to the dark components only through the gravity. Thus, the energy balance equations for these fluids may be described Cueva-Nucamendi2012 (),

d¯ρbdz−3¯H¯ρb

=

0,

(1)

d¯ρrdz−4¯H¯ρr

=

0,

(2)

d¯ρDMdz−3¯ρDM(1+z)

=

−¯Q¯H(1+z),

(3)

d¯ρDEdz−3(1+ωDE)¯ρDE(1+z)

=

+¯Q¯H(1+z),

(4)

where ¯ρb, ¯ρr, ¯ρDM and ¯ρDE are the energy densities of the baryon (b), radiation (r), DM and DE,
respectively, and ωDE=PDE/¯ρDE<0.
We also defined the critical density ρc≡3¯H2/8πG and the critical density today ρc,0≡3H20/8πG (here H0 is the current value of the Hubble parameter). Considering that A=b,r,DM,DE, the normalized densities are

¯ΩA≡¯ρAρc=¯ρA/ρc,0ρc/ρc,0=¯Ω⋆AE2,ΩA,0≡ρA,0ρc,0,

(5)

and the first Friedmann equation is given by

E2

≡

¯H2H20=8πG3H20(¯ρb+¯ρr+¯ρDM+¯ρDE),

(6)

=[Ω⋆b+Ω⋆r+Ω⋆DM+Ω⋆DE].

The following relation is valid for all time

¯Ωb+¯Ωr+¯ΩDM+¯ΩDE=1.

(7)

ii.1 Parameterizations of ¯Q and ωDE

Due to the fact that the origin and nature of the dark fluids are unknown, it is not possible to derive ¯Q from fundamental principles. However, we have the freedom of
choosing any possible form of ¯Q that satisfies Eqs. (3) and (4) simultaneously. Hence, we propose a new phenomenological form for a varying ¯Q
so that it can alleviate the coincidence problem. This coupling could be chosen proportional to ¯ρDM, to ¯H and to ¯IQ.
Therefore, ¯Q in the dark sector can be written as

¯Q≡¯H¯ρDM¯IQ,¯IQ≡2∑n=0λnTn.

(8)

Here ¯IQ measures the strength of the coupling, and was modeled as a varying function of z in terms of Chebyshev polynomials.
This polynomial base was chosen because it converges rapidly, is more stable than others and behaves well in any polynomial expansion, giving minimal errors.
The coefficients λn are free dimensionless parameters Cueva-Nucamendi2012 () and the first three Chebyshev polynomials are

T0(z)=1,T1(z)=z,T2(z)=(2z2−1).

(9)

Similarly, we propose here a new phenomenological ansatz for a time-varying ωDE and divergence-free at z→−1. Thus, we can write

ωDE(z)≡ω2+22∑m=0ωmTm2+z2,

(10)

where ω0,ω1 and ω2 are free dimensionless parameters. The polynomial (2+z2)−1 and the parameter ω2
were included conveniently to simplify the calculations. Here, ωDE behaves nearly linear at low redshift ωDE(z=0)=ω0 and
dω/dz|z=0=ω1, while for z≫1, ωDE(z)≃5ω2.

Now, we substitute Eq. (8) into Eq. (12), and then, we impose the following condition dR/dz=0
to guarantee the possibility that the coupling can solve the coincidence problem. This implies two solutions R+=R(z→∞) and R−=R(z→−1),

R+=−(1+3ωDE¯IQ),R−=0.

(13)

From Eq. (13), we note that R+ and ¯IQ are not independent but its product can be approached to the order unity. So,

¯IQR+∼−3ωDE.

(14)

In the limiting cases z→∞, z→0 and z→−1, R must be either constant or very slowly. Then, the quantities R+, R− and
the value of R today (R0) must fulfill 0≤R−<R0<1≪|R+|.

ii.3 DE models

ii.3.1 ΛCDM model

Fixing both ωDE(z)=−1 and ¯Q(z)=0 into Eqs. (1)-(4) and solving Eq. (12), we can find E2 and R

E2(z)

=[Ωb,0(1+z)3+Ωr,0(1+z)4+Ω⋆DM(z)+ΩDE,0],

R(z)

=R0(1+z)3,

Ω⋆DM(z)

=ΩDM,0(1+z)3.

(15)

ii.3.2 CPL model

Replacing both ωDE(z)=ω0+ω1[z/(1+z)], where ω0, ω1 are real parameters and ¯Q(z)=0 into Eqs. (1)-(4),
and solving Eq. (12), we obtain E2 and R,

Eqs. (8)-(12) show that from simple arguments based on the evolution of R, one can find an appropriated restriction for the coupling ¯Q between the dark components of the
universe Campo-Herrera2015 ().
On the other hand, the coincidence problem can be alleviated, if we impose that ¯Q≥0 in Eq. (11). From here, and using Eq. (12) at present time, we find

∣∣∣(¯IQ,0(1+R0)ω0+3)R0ω0∣∣∣≤3R0.

(19)

This result means that the slope of R at z=0 is more gentle than that found in the ΛCDM model Campo-Herrera2015 ().

ii.4 Crossing of ¯IQ=0 line with a coupling.

From Eqs. (8) and (9), we note that exist real values of z that leads to ¯IQ(zcrossing)=0, which
are called the redshift crossing points, zcrossing

This result depends of the choice for ¯IQ. However, the only possibility for a crossing happends when

d¯IQdz|zcrossing=±14λ2√λ12−8λ2(λ0−λ2)≠0,

(22)

From Eq. (22), we impose the following restraint to gua-rantee real values in λ1

∣λ1∣

≥

√8λ2(λ0−λ2)→(λ2≥0)∩(λ0≥λ2)

(23)

∪(λ2≤0)∩(λ0≤λ2).

In general, if λ0 and λ2 are both positive or both negative,
then d¯IQ/dz could be positive or negative. Moreover, d¯IQ/dz may be zero when λ0,
λ1, and λ2 are all zero (i.e. uncoupled DE models) or when ∣λ1∣=√8λ2(λ0−λ2). From here,
we can describe the sign of ¯Q (¯IQ).

where QA is the energy density transfer relative to UμA and FμA=a−1(0,∂ifA) is the momentum density transfer rate,
relative to UμA. Here, fA is a momentum transfer potential. We choose each UμA and the total Uμ as
Clemson2012 ()

TμAνUνA=−¯ρAUνA,TμνUν=−¯ρUμ,

(29)

Thus, the total energy-frame is defined as

(p+ρ)v=∑A(¯ρA+¯pA)vA,

(30)

where v is the total energy-frame velocity potential. From Eqs. (25) and (28) obtain

QA0=−a[¯QA(1+ϕ)+δQA],QAi=a∂k[fA+¯QAv].

(31)

The perturbed energy transfer QA0 includes a metric perturbation term ¯QAϕ and a perturbation δQA. In addition, we stress that
the perturbed momentum transfer QAi is made up of two parts: the momentum transfer potential ¯QAv and fA.
On the other hand, the physical sound-speed csA of a fluid or scalar field A is defined by c2sA≡δPA/δρA∣rf in the A rest-frame (rf), and the adiabatic
sound-speed csa can be defined as c2aA≡P′A/ρ′A=ωA+(ω′A/¯ρ′A)¯ρAvaliviita2008 ().
For the adiabatic DM fluid, c2sDM=c2aDM=ωDM=0. By contrast, the DE fluid is non-adiabatic and to avoid any unphysical instability, c2sDE
should be taken as a real and positive parameter. A common choice (and the one we make here) is to take c2sDE=1valiviita2008 (); Clemson2012 ().
Defining the density contrast as δA≡δρA/¯ρA, we can find equations for the density perturbations δA and
the velocity perturbations θAvaliviita2008 (); Clemson2012 (),

δ′A

=−3¯H(c2sA−ωA)δA−9¯H2(1+ωA)(c2sA−c2aA)θAk2

−(1+ωA)θA+3(1+ωA)ψ′A+a¯ρA(−¯QAδA+δQA)

+a¯QA¯ρA[ϕ+3¯H(c2sA−c2aA)θAk2],

(32)

θ′A

=−¯H(1−3c2sA)θA+c2sA(1+ωA)k2δA−a(1+ωA)¯ρAk2fA

+a¯QA(1+ωA)¯ρA[θ−(1+c2sA)θA]+k2ϕ,

(33)

The curvature perturbations on constant-ρA surfaces and the total curvature perturbation are given by

ζA=−ψ−¯HδA¯ρ′A,ζ=∑A¯ρ′A¯ρζA,

(34)

Then, we need to specify a covariant form of QμA and fA in the dark sectors valiviita2008 (); Clemson2012 ().
For fA, the simplest physical choice is that there is no momentum transfer in the rest-frame of either DM or DE valiviita2008 (). This leads to
two cases

For both cases Qμ∥UμDM or Qμ∥UμDE cases, Eq. (III) does not change, but Eq. (III) is different in both cases.
In this article, we focus only on the Qμ∥UμDM case. Besides, assuming that ¯Q depends only on the cosmic time through the
global expansion rate, then a possible choice can be

In a forthcoming article we will extend our study, by considering other relations between δIQ, δDM and δH. It is beyond the scope of the present paper.
Considering that c2sDM=0, c2sDE=1 and using the above Eqs into Eqs. (III) and
(III), we find valiviita2008 (); Mas ():